Further simplification of the super-Hamiltonian constraint of General Relativity, and a reformulation of the Wheeler-DeWitt Equation

The super-Hamiltonian of four-dimensional gravity as simplified by Ashtekar through the use of gauge potential and densitized triad variables can furthermore be succinctly expressed as a Poisson bracket between fundamental invariants. Even when a cosmological constant is present, the constraint is equivalent to the vanishing of the Poisson Bracket between the volume element and a combination of the integral of the trace of the extrinsic curvature and the Chern-Simons functional. This observation naturally suggests a reformulation of non-perturbative quantum gravity wherein the Wheeler-DeWitt Equation is reduced to the requirement of the vanishing of the expectation value of the corresponding commutator. Remarkably, this formulation singles out spin network states as explicit realizations of the physical states. Moreover, by requiring physical states to be simultaneous eigenstates of the commuting operators, the formulation also yields a Schrodinger Equation with "intrinsic-time development".